NUMERICAL EXPERIMENTS WITH A FIVE-LEVEL GLOBAL ATMOSPHERIC PREDICTION MODEL USING A STAGGERED, SPHERICAL, SIGMA COORDINATE SYSTEM William Theodore Eli as m Library Naval Postgraduate Scfiod Monterey, California 93940 \S iJ t %^ 'l^ a t^ %3 SI i im W^lflWUla ORlepey, California 1^ ii NUMERICAL GLOBAL USING A EXPERIMENTS ATMOSPHERIC STAGGERED , COORDINATE WITH A FIVE-LEVEL PREDICTION MODEL SPHERICAL, SIGMA SYSTEM by William Theo dore Elias George Haltiner Th esis Advisors: Roger Williams 1 March 1973 T ■'^'? Appnjovzd. {^OJi pabtlc n.zt^ja^e,; dUt/uhvution urJianittd. Numerical Experiments with a Five-Level Global Atmospheric Prediction Model Using a Staggered, Spherical, Sigma Coordinate System by William Theodore Elias Lieutenant, United States Navy B.S., California Polytechnic College at Pomona, I968 Submitted in partial fulfillment of the requirements for the degree of MASTER OP SCIENCE IN METEOROLOGY from the NAVAL POSTGRADUATE SCHOOL March 1973 Library J^aval Postgraduate School 'Monterey. California 93940 ABSTRACT Three cases of analytic data and one case of real data were numerically integrated using a 5-level baroclinic primitive equations model of the general circulation. Experiments were performed using initial winds derived from the linear balance equation and also winds derived analyti- cally. The feasibility of using the linear balance equation to initialize the wind field v/as examined. In all cases, the forecasts remained meteorological and reasonably well-behaved. Nevertheless, the forecasts derived from initial winds generated by the linear balance equation excited large, operationally-undesirable inertial-gravity waves, while the forecasts from analytically determined Initial winds remained virtually free of such small scale "noise". TABLE OF CONTENTS I. INTRODUCTION 11 II. BAROCLINIC PRIMITIVE EQUATION MODEL 12 A. PRIMITIVE EQUATIONS 12 B. GRID m C. VERTICAL LAYERING lH D. TIME DIFFERENCING 17 III. INITIAL CONDITIONS 19 A. ANALYTIC WINDS 22 B. DERIVED WINDS 23 IV. FOURIER WAVE ANALYSIS METHOD 25 V. RESULTS 26 VI. CONCLUSIONS 5.6 LIST OF REFERENCES 58 INITIAL DISTRIBUTION LIST 60 FORM DD 1473 64 LIST OF CHARTS A. Initial Surface Pressure Analysis (Experiment IV) ^0 B. Verifying Surface Pressure Analysis (Experiment IV) ^1 C. 12-Hour Forecast Using FNWC Model (Experiment IV) 42 D. 12-Hour Forecast Using Global Model (Experiment IV) ^3 E. Initial Surface Pressure Analysis (Experiment I) 44 F. 12-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment I) 45 G. 24-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment I) 46 H. 36-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment I) .47 I. Initial Surface Pressure Analysis (Experiment II) 48 J. 12-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment II) 49 K. 24-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment II) 50 L. 36-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment II) 51 M. Initial Surface Pressure Analysis (Experiment III) 52 N. 6-Hour and 12-Hour Surface Pressure Forecast Using Analytic Winds (Experiment III) 53 0. 18-Hour and 24-Hour Surface Pressure Forecast Using Analytic Winds (Experiment III) 54 P. 30-Hour and 36-Hour Surface Pressure Forecast Using Analytic Winds (Experiment III) 55 LIST OF FIGURES 1. Location of Variables 15 2. Vertical Layering l6 3. Plot of Terrain Pressure Versus Forecast Hour at 0°N 15°E (Experiment IV) 3^ k. Plot of Terrain Pressure Versus Forecast Hour at hO°n 15° E (Experiment IV) 35 5. Plot of Terrain Pressure Versus Forecast Hour at 85°N 15°E (Experiment IV) 36 6. Plot of Terrain Pressure Versus Forecast Hour at 0°N 15°E for "Analytic" Winds (Experiment II) 37 7. Plot of Terrain Pressure Versus Forecast Hour at 40°N 15°E for "Analytic" Winds (Experiment II) 38 8. Plot of Terrain Pressure Versus Forecast Hour at 85°N 15°E for "Analytic" Winds (Experiment II) 39 LIST OF TABLES I. Wave Amplitudes and Mean Heights For Selected Latitutdes and Forecast Time 31 II. Computed Phase Speeds at Selected Latitudes Along with Method of Balancing and the Time Required To Make A 36-Hour Forecast For Each Experiment 32 III. Comparison of 12-Hour Forecast Pressures of the FNWC Model and the Global Model Against the Verifying Pressure Analysis 33 LIST OF SYMBOLS AND ABBREVIATIONS A - Arbitrary constant in the stream function A - Arbitrary constant for the Fourier series ^ cosine terms a - Earth's radius B - Arbitrary constant in the stream function B - Arbitrary constants for the Fourier series m . , sine terms CDC - Control Data Corporation C - Arbitrary constants for the Fourier series ^ combined terms Op - Specific heat for dry air at constant pressure c - Wave speed D - Lateral diffusion for the quantity indicated by the subscript D„ - Lateral diffusion of heat D - Lateral diffusion of specific humidity D - Lateral diffusion of momentum m FNWC - Fleet Numerical Weather Central F •• - Frictional stress f - Coriolis parameter f - Coriolis parameter at 45° north g - Acceleration of gravity H - Diabatic heating 1 - Grid index in the x-direction (east-west) J - Grid index in the y-direction (north-south) M - Stands for <}), T, q, and w m - Wave number mb - Millibars NPS - Naval Postgraduate School NACA - National Advisory Committee on Aeronautics N - Wave num;ber plus one - m + 1 n - Degree of the Legrendre function P - Pressure P ' - Legrendre function of order m and degree n Q - Moisture source/sink term q - Specific humidity R - Specific gas constant for dry air t - Time u - Zonal Wind ^ - Horizontal vector velocity - (u,v) V - Meridional wind W - Stands for u and v w - Measure of vertical velocity, positive upward w = - g = - da dt At - Time increment Ax - Distance increment in the x-direction - a AX cos a - Specific volume 6 - Phase angle for wave number m m ^ 6 - Latitude, positive northward from south pole AG - Distance increment in the latitudinal direction X - Longitude, positive eastward from Greenwich 8 AX - Distance increment in longitudinal direction V - Angular v/ave velocity TT - Terrain pressure parameter a - Dimensionless vertical coordinate, <_ a £ 1, increasing downward • dcT G - Measure of vertical velocity - ^, -w <}) - Geopotential ip - Stream function V - Del operator (horizontal) 2 V - Laplacian operator (horizontal) ACKNOVJLEDGEKENTS The author wishes to express his thanks to Dr. G. J. Haltiner for his encouragement to undertake this project. Dr. R. T. Williams for his patient guidance without which this project would never have been completed, and Dr. P. J. Winninghoff for providing the original program. In addition, the computer support of FNWC, under the command of Captain W. S. Houston Jr. USN, was both essential and greatly appreciated. The author would also like to express his thanks to several members of Captain Houston's staff, particularly Commander Celia L. Barteau, USN, and Mr. Leo Clarke who provided coordination of computer services. 10 I. INTRODUCTION In the field of operational numerical weather prediction, the tendency, in I'ecent years, has been toward the develop- ment of sophisticated global prediction models. This has been made possible by the rapid growth of computing capacity and developments connected with general circulation research. The purpose of this study vias to examine a baroclinic primitive equation model using a global, staggered, spherical, Sigma coordinate system which could be used on an operational day to day basis by the United States Navy. In order to do this, an input procedure was developed to allow a real time Initialization from FNWC analysis fields. A set of analytic fields using an analytic spherical harm.onic stream function, first presented by Neamtan (19^6), was also developed for use as a controlled set of initial conditions. By using these analytic cases, errors in real data collection, analysis and initialization, which are inevitable in pract: - cal meteorology, were avoided. The use of an analytic case also allowed the control of temperature and moisture distri- bution, predominant wave number, phase speed and wave amplitude. The objective was to isolate and correct problems in the model by using the controlled analytic input with the ultimate aim of improving and extending the Navy's overall capability to predict weather phenomena on a global scale. 11 II. BAROCLINIC PRIMITIVE EQUATION MODEL The governing differential equations, written in vector form, are similar to sets used by Smagorinsky et al. (1965), Arakawa et al. (1969) and Kesel and Winninghoff (1972). The integrations were carried out on a global, spherical, staggered, sigma. coordinate system using the conservative- type difference equations based on the work of Arakawa (1966). The complete set of finite difference equations are given in Kesel and Winninghoff (1972) for the non-staggered square grid. The Arakawa type spatial differencing was used to eliminate the spurious energy growth which can occur with the more conventional finite difference approximations to the nonlinear advection terms. A. PRIMITIVE EQUATIONS The vector equation of horizontal motion in the sigma (a) coordinate system for this model is: ^(ttV) + (V-7tV)V + 7T ^(aV) + f(KX7TV) + TrVcf) + QTraVTT = F + D„ d u do in The Thermodynamic energy equation is: iTr(^T) + V-(TrTV) + it |-(Ta) - — (a|^ + aV-VTT + -no) = ttH + Dn, oil do Cpdo J. 12 The mass continuity equation is: |i+ vc.^) + .||= The moisture continuity equation is : |^(7Tq) + V.(7TcV) + 7T l^-Cqa) = ttQ + D^ These equations are supplemented by the hydrostatic equation ?^ + TTa = 9a the equation of state a = RT/P and the dimensionless vertical coordinate a = P/tt In the above equations, u is the zonal wind component; Vj the meridional wind component; tt, the terrain pressure; a the vertical-velocity measure; T, the temperature; f, the Coriolis parameter; 6, the geopotential; q, the specific humidity; H, the diabatic heating; F, the frictional stress; 13 D , the lateral diffusion of momentum: D^, the lateral diffusion of heat; D , the lateral diffusion of moisture: ' q ' Q, the moisture source/sink term and P, the pressure. Furthermore, the formulations for frictional stress (F), lateral diffusion (D), heating (H) and moisture (Q) are identical to those described in Kesel and Winninghoff (1972). B. GRID The spatial finite differencing was performed on a staggered, spherical grid. The geopotential ((})), tempera- ture (T), specific humidity (q) and the vertical-velocity measure (w = - a) were carried at the poles. The longitudinal and latitudinal grid increments were both five degrees. This gives 2520 points over the globe, with (4',T,q,w) and (u,v) carried at 1260 points each (See Fig. 1). When (j>,T,q or w variables are needed at (u,v) points (e.g. T in the friction term and w in the vertical advection of momentum term) , they are defined as the average of the 4 values at the surrounding (4'>T,q,w) points. C . VERTICAL LAYERING The present model divides the atmosphere into five layers, as sketched in Figure 2. The basic variables of the model are carried at the center of each layer. Addi- tional variables and conditions are specified at the interface between layers, as well as the upper and lower boundaries . ih M (North Pole) W M W M .... M M V/ M W .... W W M W M .... M W M W M W + J- W M W M • • • • M (South Pole) FIGURE 1. LOCATION OF VARIABLES M stands for 4), T, q and w variables W stands for u and v variables 15 M Level Computed Variables (Notational (at each level) Subscript) Sigma Pressure (a) Q Q w = UPPER BOUNDARY ^ q 1 X IT .u^v^T^^ 5 . 2 X 7T ^ ^ 0.2 3 X IT .H-.Y-.T^^ i . 3 .4 X TT "^ ^ 0.4 .5 X TT H-LY^T^*-.q 3 0^5 .6 X 7T "^ ? 0.6 .7 X ^ HiY^T^^aS 2 Q^^ 8 X TT "^ 1 ■ 0.8 9 X TT "^xIjilA^S. I 0.9 ] w 1 ^xijilA^s. w = 1 LOWER BOUNDARY FIGURE 2. VERTICAL LAYERING In the above figure, sigma (a) is the dimensionless vertical coordinate, u, is the zonal wind component, V is the meridional wind component, q is the specific humidity, <{) is the geopotential , it is the terrain , pressure and w is the vertical-velocity measure (-a). o 16 Integrations are performed in the Phillip's (1957) Sigma (a) coordinate system, where pressure, P, is normalized with the underlying terrain pressure, tt. The dimensionless vertical coordinate, sigma (a), is defined as a = P/tt It follows rhat 0=1 at the earth's surface and a = at the top of the atmosphere. The zonal wind component (u), the meridional wind compo- nent (v) , the temperature (T) and geopotential ((j)) are carried at sigma levels .9> .7, .5, .3 and .1. The specific humidity (q) is carried at sigma levels .9, .7 and .5. The vertical velocity measure, w = -a, is calculated diagnosti- cally from the continuity equation for the layer interfaces. The vertical-velocity measure (w) is carried at sigma lev is .8, .6, .4 and .2. D. TIME DIFFERENCING The two finite difference techniques that were used to step forward in time are the centered (leapfrog) difference schem.e and the Matsuno (Euler backward) difference scheme. The finite difference equation for the centered technique is: pt+1 ^ pt-1 ^ 2At ^^' 8t 17 Since the form has three time levels, it has both a physical and a computational mode (Haltiner, 1971). This computational mode, which changes phase at every time step, causes the solu- tions at adjacent time steps to become decoupled. Further- more, the centered scheme is not feasible for the first time step. Therefore, the Matsuno forv/ard time differencing scheme is used for the initial time step of each 6-hour integration cycle. This time step not only reduces solution separation but also selectively dampens high frequency waves (Haltiner, 1971). The finite difference equation for the Matsuno scheme is: F* = F^ + At ^^ dt pt+l = pt ^ ^^ |Fi dt A time step of ten minutes was used for all experiments. This large time step is comiputationally stable when the Arakawa technique (Langlois and Kwok, I969) of averaging quantities involved in the longitudinal derivatives is use u Without this averaging technique, the von Neumann linear computational stability criterion (Haltiner, 1971) would require a 2.5 minute time step because the longitudinal grid distance at 85° north and south is only 47 kilometers. No averaging was done equatorward of 60° north and south. It should be noted at this time that lateral diffusion, friction, convective adjustment, convective condensation and large scale condensation are computed at every time step while heating is computed every 6 time steps (Kesel and Winninghoff, 1972). 18 III. INITIAL CONDITIONS Both real and analytic data were used for initialization. The data was produced external to the main program on a Northern Hemispheric FNWC 63 x 63 grid. The analytic data with known phase propagation properties was produced following the work of Neamtan (19^6), Gates (1962) and Heburn (1972). The real data was read from FNWC tapes. These fields were analyzed by the FNWC objective schemes. The main program was used to interpolate the data so that values at 5° latitude-longitude intersections were available over the Northern Hemisphere. The data was then reflected into the Southern Hemisphere. This procedure allows for an excellent test of the program. The model begins with fields which are mirror images of each other and they should remain so. In all cases, the input data consisted of the temperature analyses for the Northern Hemisphere at 12 constant pressure levels distributed from 1000 to 50 mb, height analyses at 10 of these levels, moisture analyses at 4 levels from the surface to 500 mb, sea level pressure and sea surface temperature. Monthly mean surface temperature fields were used to derive an albedo field (Dickson and Posey, I967). The terrain height, in the real data case, was that used by the current FNWC model. The terrain height, in the analytic data cases, was set to zero. 19 In order to Isolate and evaluate certain numerical errors, it was decided to use analytic initial conditions for the stream function il^, from which the true solution of the complete vorticity equation in two dimensions could be obtained. With the assumption of nondivergent horizontal flow, harmonic wave solutions of the complete vorticity equation have been obtained for the sphere by Neamtan (19^6). These solutions yield the velocity of propagation of the waves. The solution for the ^ field was found to be i|; = A sin(mA-vt) P^^(sin G) - B r^ sin 6 + C Pj^(sln 6) (1) where A, B and C are constants to be determined, r is the radius of the earth, m is the hemispheric wave number, v/m is the angular phase speed of the wave disturbance (radians / second) , P denotes the Legendre polynomial and P ^ represents the associated Legendre function. If we choose C = (Gates, 1962) and n = m+1 [Buringt n and Torrance (1936), Kreyszig (1962)] equation (1) reduces to ip = A sin(mX-vt)(2N!/2\! ) sin (f) (cos(j)) - B r^ sin <|) . (2) The constant A is arbitrary and proportional to the wave amplitude. But it was shown by Haurwitz (19^0) that the solution obtained for i> implies the existence of a velocity distribution over the sphere such that the angular velocity of the westerly current is made up of the sum of three terms. 20 The first term, B, is constant over the sphere; the second term varies along a meridian but is constant round a circle of latitude; and the third term represents a harmonic wave which is propagated zonally with a constant angular velocity, v/m, which is given by the formula / □ n(n+l) - 2 2fi f^. v/m = B — ^^ / , ., \ / . -, V (3) nCn+1) n(n+l) ^-^^ where fi is the earth's rotation speed. A reasonable meteorological pattern was obtained by 2 —1 choosing A as 1000 m sec , m as 6 and v/m as -13°, 0° and +12° longitude per day respectively. The ip field thus obtained was used to determine the initial fields. The geopotential fields were derived by solution of the linear balance equation V^ (j) = [f V^4; + Vi|;- A f] (^) over the entire Northern Hemisphere. Equation (4) was solved using the over-relaxation iterative technique with a relaxation tolerance of one meter (Haltiner, 1971) • The temperatures generated for the analytic cases were constant over each pressure surface and were consistent with the National Advisory Committee for Aeronautics (NACA) standard atmosphere (Haltiner and Martin, 1957). The eleva- tion 10 ,769 meters represents the tropopause in this 21 atmosphere. The following relations were used in calculating the analytic temperatures. Tp(<^K) = 288 - 0.0065 Z Z <_ 10,769 meters Tp(°K) =218 • Z 1 10,769 meters Zp(m) = i|4,308 [1 - (^^l_^)0-'-90233 ^ 1 10,769 meters Zp(m) = 10,769 + 6381.6 ln( ^^ p ^^ ) Z i 10,769 meters In the foregoing equations, the pressure is in millibars and the height Z, is in meters. In addition, the specific humidity (q) at the lowest three levels was set to zero and the forecast procedure excluded friction, convection adjust- ment, large scale convective condensation, large scale condensation, heating and terrain influences. The initial wind fields in the analytic cases were obtained in two ways. Winds obtained analytically from tl 3 original ^ fields were called "analytic winds." Winds obtained from the linear balance equation were called "derived winds." It should be noted that "derived winds" v;ere used for the real data case. A. ANALYTIC WINDS The initial wind fields for the "analytic wind" cases were derived directly by differentiation of the initial 22 rl> field (Neamtan, 19^6). The formulas for the non-divergent wind components are u = - - (a sin(mX-vt)(-^^)[(cos 6)"^- 6 (sin e )^ (cos 6)^] - B r^ cos e I (5) V = r A ( . ' ) sin 6 (cos G) m cos(mX-vt) (6) where a is the earth's radius, A is an arbitrary constant that is proportional to the wave amplitude, 6 is the latitude, V is the angular velocity of the wave, A is the longitude, m is the wave number and N = m+1. B. DERIVED WINDS The "derived wind" field was obtained from a ^ field using the finite difference expressions 1 ^\b ^ = " i Ae and 1 ^± a cos 6 AX where a is the earth's radius, AS is the latitudinal distance increment and AX is the longitudinal distance increment. The rp and 4) fields were obtained in two parts 23 depending on latitude. Poleward of 25° north and south , the initial* geopotential field, say <t> , was retained and the ^ field was obtained by solution of the linear balance equation V^ij; + Vi|^-Vf/f = V^4) /f . Equatorward of 2 5° north and south , the field was deter- mined using ^ = <J>o/f where f is a mean coriolis parameter. Then a new geopoten- tial field (4)) was obtained by solution of the linear balance equation V^ = (fV^ip + Vi^-Vf) At 25° north and south, ^ and <t> were a combination of half the i> and (p from the poleward case and half the ip and (j) fr -m the equatorward case. It was hoped that this would reduce the amplitude of the high frequency oscillations in the tropics (Winninghoff , 1971). By initial geopotential field is meant either the FNWC objective analysis of <j) or a simulated <t> field-obtained by solution of Equation (h) using the analytic function form of ^ (Equation (1) ) . 2k IV. WAVE ANALYSIS METHOD A Fourier series was determined at each five degrees of latitude around the latitude circle. The employment of such a series, known as harmonic analysis, is extensively used in the study of observational data (Jeffreys and Jeffreys, 1956). With this technique, the phase angles and amplitudes of each wave number around a latitude circle can be calcu- lated. A Fourier series can be expressed as follows (Heburn 1972): F(x) = A^ + E (A cos mx + B sin mx) o ^ n^ m m = C + I (C cos (mx - 5m) o ^ ni m where B A c = m _ m m sin(6m) cos (6m) and 1 B 6^ = tan"\/^). The first three experiments involved an input stream function of wave number six. The values of primary interest, there- fore, were 6r and Cr which are the phase angle and amplitude of v;ave number six. Other phase angles and amplitudes were extracted and examined, especially in the real data case. 25 V. RESULTS Four experiments will be presented in this section. The first three of these experiments were performed with analytic fields. These analytic fields were derived from a stream function with a wave number of 6 and arbitrary constant, 2 -1 A, set equal to 1000 m sec . The vertical temperature structure of these fields was determined by the NACA stan- dard atm.osphere. Experiments I and II consisted of two 36-hour forecasts each. The first forecast used "analytic" winds and the second forecast "derived" winds. Experiment III consisted of one 36-hour forecast using "analytic" winds. Experiment IV was performed with FNWC analyses and consisted of one 36-hour forecast using "derived" winds. Experiment I. The analytic geopotential field used in this experiment was derived from a stream function with a wave number of 6 and phase speed of -13° longitude per day. T .e initial surface pressure analysis is shown on chart E. Tie forecast surface pressure fields using both "analytic" and "derived" winds can be found on charts F-H. Experiment II . The analytic geopotential field used in this experiment was derived from a stream function with a wave number of 6 and phase speed of 0° longitude per day. The initial surface pressure analysis is shown on chart I. The forecast surface pressure fields using both "analytic" and "derived" winds can be found on charts J-L. 26 Table I compares the wave amplitudes and mean heights of the 36-hour forecast pressure field for experiment II (analytic winds) with those of the initial pressure field at selected latitudes. The amplitudes and mean heights in this experiment decreased at latitudes equatorward of 30° north (south) and increased poleward of 30° north (south). These amplitude variations are believed to be caused by nonlinear effects. In addition, Figures 6-8 are plots of terrain pressure versus forecast hour at selected latitudes. These terrain pressures are plotted every hour out to 36 hours. They show small amplitude internal gravity waves with periods ranging from 6-12 hours. These oscillations are believed to be due to the geostrophic adjustment mechanism and are peculiar to the type of analytic field used. Experiment III . The analytic field used in this experime* t was derived from a stream function with a wave number of ' and a phase speed of 12° longitude per day. The initial surface pressure analysis is shown on chart M. The fore- cast surface pressure fields using "analytic" winds only can be found on charts N-P. Experiment IV. This experiment was performed using FNWC objective analyses for OOZ February 8, 1973. The initial surface pressure analysis is shown on chart A. The surface pressure analysis for 12Z February 8, 1973 is shown on chart B. The 12-hour forecast made by the current F^^WC 27 model is shown on chart C. The 12-hour forecast made by the global m.odel is shown on chart D. It should be noted that Experiment IV included terrain, frictional stress, heating and moisture effects as discussed in Kesel and Winninghoff (1972). Table II summarizes the results of all experiments performed. The analytic phase speeds are compared to the actual phase speeds for all experiments using analytic data. The table also includes A which is proportional to the amplitude of the analytic wave, B which is a function of the phase speed and wave number, the method of balancing and the time required to make a 36-hour forecast. The forecast fields in all the experiments using analytic data showed a considerable tilt backward at high latitudes in the phase propagation of the wave. This was to be ex- pected since the Arakawa averaging technique tends to smcth the gradients at high latitudes. In addition, the Arakaw . technique gives an effective Ax which is comparable to tY it at low latitudes, thus as the wavelength decreased toward the poles the phase speed also decreased (Gates, 1959). The result of this differential movement, which was more pronounced in the "derived" wind cases, was the forma- tion of closed highs and lows at the higher latitudes which propagated equatorward. This distortion was aggravated by nonlinear effects introduced after the field ceased to be harmonic in the longitudinal direction. 28 Furthermore, in all the experiments using the "derived" wind, high frequency inert ial gravity vraves were generated due to the initial im.balance between the mass and wind fields in the tropics. These gravity waves caused a rapid deterioration of the harmonic six-wave pattern and added appreciably to the problem at the poles. The shear which developed at 25'^ north (south) was attributed to the method used to blend the v/inds between the tropics and mid-latitudes. The larger error in the phase speeds calculated for the "derived" wind cases were also attributed to this initialization Since these experim.ents were performed using a multi- level primitive equation model which allows divergence. Equation (3) was satisfied only approximately. Also, Rossby (1939) has shown that the presence of divergence in a barotropic atmosphere will slow the rate of wave propaga- tion, especially for small values of wave number. It was not surprising, therefore, that the actual phase speed Wc.: always less than expected. Finally, a thorough examination was made of Experiment IV to determine the general quality of the program, that is, if the inertial-gravity motions were being controlled realistically so that the model produces consistent meteor- ological appearing results. Figures 3-5 give the surface pressure oscillations at each hour of a 36-hour forecast for selected latitudes using the full global model. In addition. Table III compares forecast pressures of the global model to both the verifying analysis and forecasts 29 made by the FNV/C model for selected lows and highs. In interpreting these results one should keep in mind the crude initialization. Also, for those not accustomed to seeing values of terrain pressure at each hour in a primitive equation model, these oscillations, although larger than normal at these points, are by no means unusual for such a poorly initialized integration. There is no artificial smoothing done and recall that the run begins with a linear balance wind. In addition, there are none of the various devices which operational experience has shown is necessary to give an acceptable short term product, i.e., not calling on convective adjustment for the first several time steps. Nevertheless, these results are stable and retain the dominant meteorological scale without difficulty. 30 TABLE I Latitudes Amplitude of Wave #6 (Cg) Mean Pressure (C-, ) Equator 10° N 20° N 30° N ^0° N 50° N 60° N 70° N 80° N Initial Pressure Field . 5. .4 18. .2 29. .1 30. .2 21. .5 10. .1 2. .7 1 .31 .01 36-Hour Forecast Initial 36-Kour Forecast Pressure Pressure Pressure Field 11.0 24.0 30.2 25.5 15.0 5.8 1.5 .37 Field Field 1051 1040 1035 1034 1016 1015 995 995 973 977 953 956 935 937 922 924 914 919 Note: Data is from Experiment II (Analytic Winds) TABLE I. Wave Amplitude and Mean Heights for Selected Latitudes and Forecast Time. 31 '^' o in o o IS o o CO s CQ rH W (U O CD Cd <U iH ^S CO CO CVi rH C\J cr\ H 1 1 1 1 OJ I CO I CO o vo o o o CO 1 in I CO + o VD o o o I in o o o •H •d •H 4-> (1) +3 >s W > CO >5 CO rH T? •H 'O r^ 73 Cd C ^ c Cd c C -H 0) ^H C -H < rs q:s <:s vo m i CO vo o •d <u > CO •H 73 U C Q) "H Q ^ VO t-- VO 1 C— VO 1 • X o • X o CM rH rvj iH O o O o O o rH t-i 4J c B O •H (1> 6 o -P CO •H O o O ;^ o T-l w <D in M (U • CI, X w vo Cd nH o cd y <D -p 0) CO Cm S H . •\ o x: o rH ^ CO o Td cd t~ CO cd <D •H o + d) CO J-i •H O CD 4^ (D P 4^ x: :3 T? o Cd -P fn rH t3 inclu as F erpol 0) O S "M £ +3 rH cu ^ -p 4-> CO + 5 +J o to • 5 »» •H Cd 0) fn CO CO bO S^H m (D 0) •H C O + •H omput out in CO >3 O fin rH < U ^ o u CD CO O Hf ^3 1 :3 vo 4J CO •H +j cd cd in in kJ o CO ^ r-i rH ected to Ma O rH •H •CJ d) "TZi +5 0) zn <D >J CO > CO u iH 'O •H Td ^ -H Cd c U C ' :3 C -H <U -H D^ < rs Q :s ■ (U ^^ QC - <U .6 vo 1 ; -H Eh J vo CO Td Cvj 1 1 1 cd c • X o 1 x: cd in rH bO 13 C o 1 <U -H o 4^ O o i :3 c iH Comp Bala H H M H OJ H + M H M > M H EH 32 TABLE III Location of Pressure System on Initial Surface Analysis 65°N 135°W 70°N 15°E 65°N 25°W 65°N 70°W 50°N 165°E 55°N 115°E 55°N 45°E 50°N 55°W 50°N 100°W 50°N 160°W 35°N 175°W 30°N 20°W 35°N 55°W Initial Surface Verifying Pressure Analysis Surface Pressure OOZ 8 Feb 1973 Analysis 12Z 8 Feb 1973 1042(H) 971(L) 991(L) 999(L) 966(L) 10il9(H) .97ML) 1031(H) 1037(H) 1004(L) 1030(H) 1006(L) 1012(L) 10^1 975 990 992 962 1046 976 1030 1037 1009 1030 1009 1006 FNWC Global 12-hour 12 -hour Surface Surface Pressure Pressure Analysis Analysis 1040 1042 976 973 989 989 992 992 959 962 1043 1052 975 975 1029 1036 1037 1040 1004 1005 1029 1031 1007 1003 1007 1010 TABLE III. Comparison of 12-Hour Forecast Pressures of the FNWC Model and the Global Model Against the Verifying Pressure Analysis (Experiment IV) 33 w o <L) U O O O O o o W o LTv 4^ cd o -p CO a o ^ o w 13 CO fn <U > CO JO Q) u c •H rt > ^H M <D -(J o u ^ 0) O Cu rH X en faO •H (in O OJ o in r-{ o o O O O on o o aangsaaj ufeaasj, 34 o <^ 2: o o w o 4^ cd o w (0 o u o m :3 (0 f^ > CO CO <D U cti U 0) Eh O O > 4J C 6 •H 0) bO •H fin O LTv o in m CTv sanssaaj uT^adaj, 35 u o (m s o . in CO W o LP\ iH . -P CCJ ^ :3 o ffi -p w a o <D ?H O -p fo w cd CO o :3 <D to U ^ O <u ptH > Cm <D O Jh :3 U CO :3 CO o (U w -■* • '4 • -:> . - M ) +^ H C <U <M e O -H Jh +^ <U O ft iH X P^ H in faO •H o o in o o o o o in o in aanssaaj ufeaaaj, 36 m O o o W o o in m rH Cd o LTv C\J -p w cd o (D 4J O W Ph (ti o O w . C\J CD 13^ ^ CO CO O U ^ fc CD C > ^ <;h :s o cu ^H ^ -H in :3 CO 4^ <H o CO >s K rain Pre II (Anal o f^ iH (D 4-- Eh C CU <M 6 O -H U O D. iH X P-i W 0) hO •H O vo o o in o o o iH O o rH o OJ o aj:nss8J.<j uif^aaaj, 37 CO ti o 0) u o Ph <M o u o o <M o O .=r W o in d U o ■P CO cd o (D U O w > u o 13 -H CO 4-5 CO >s 0) rH U cd < •H cd M ?^ M U d) 4J H C 0) e o 0^ w o hO •H o CVI o o o o o o O a<inss8JcI u"feaaaj, 38 u o Cm O in CO o un iH cd o -p CO cd o <D U O w :=s^ w w 0) C > nH o 0) to 4J <U rH u a CM c < cd H ^ M EH O •4-3 c p. iH X 4^ O CO bO •H ON o O CO O CVJ o cr\ rH O 0\ CT\ aanssaaj ufBadaj, 39 f mi m FEB 73-.ps. 4^95SFC\flNflL V/iGtlJBPL MGDlL tt' lOtBIC* rffl'^F CEhTffPt u. 5. I*'* CHART A. Initial Surface Pressure Analysis (Experiment IV) 40 •-.^ -v^. ■ ^ "^ '^ ^^^^^^ .--- \s«» 4- "/■■ <_ ■'■•■■""■■\ 12Z 08 FEB 75-FS/ 4760 REPORTS ■,1.5377.3' '■"^ 5BG "■■- ..J_ /■' \ -■"" ^- ' aci' luwia*. «c«'i<» ctn'iio. u. s. ww.^ 1 CHART B. Verifying Surface Pressure Analysis (Experiment IV) ^1 CHART C. 12-Hour Forecast Using FNWC Model (Experiment IV) 42 g 00Z 08 FEB 75-.P.S \.p HOlilR PROG y/^CLiD^pL'^MODEL I FLU1 itJfWlOl tCiiTO ttH'W*, u. 5. t«»ir» CHART D. 12-Hour Forecast Using Global Model (Experiment IV) 43 RNflLYTlC PSrC'flNfiL V,:£l:OBPL MODEL ti.',»c»tMC* rfai<^ ahiwi u. s- (*><< CHART E. Initial Surface Pressure Analysis (Experiment I) l\l\ 1 2 HOUR FROG V./Gt:1Jj3flL 110DEL fltE' MJtltaiCflt UtS'rttR CENTPiH. U. S. N6 Derived Winds Analytic Winds j2 HOUR PROG v;/GLljppC "MODEL fUll HJ^f>K« MCfllf^Jl aNlRfll U. S. hP'.l CHART P. 12-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment I) 45 Analytic Winds H HOUR PROG V/CLOBaL- MODEL fLttl MjftRICflL UtMTHCR aSTFSl U. S. JlHV Derived Winds in HOUR PROG V/.'-GtOBPL -flODPL flt£' HUftflCfil g£flI»tR aWWl U. S. Nl!»» CHART G. 24-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment I) 46 35 HOLiFi PROG y/:-GLliUHL MOULL niL^ Wjttfli.Hl ^H"h».« (.iMt-M', t.'. S. ffl'i Derived Winds A ' * '"' .V- ' "-' . . Analytic Winds I 36 HOUR PROG V'/XLODfiL MODEL 1 fLtE.1 NurtPICK itfP'HtB CtNTPi^L U. S. W=V' CHART H. 36-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment I) ^7 CHART I. Initial Surface Pressure Analysis (Experiment II) 48 Analytic Winds 12 HOUR PROG v'/.'-GLOf.iHL fiuDLL fltf-I HiUKf- l>:ni. UCMiHLh CC'j'finL u. s. Derived Winds L._. ; i: HOUR FROG '•/..•-'■Gl-OieHL MODE L fiU' iMtniCn uts'BEs ammn u. s. m>ji CHART J. 12-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment II) ^9 Derived Winds Analytic Winds CHART K. 24-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment II) 50 Derived Winds Analytic Winds i 36 HOUR FROG V.{i:iLl3BPL MODtL CHART L. 36-Hour Surface Pressure Forecast Using Analytic and Derived Winds (Experiment (II) 51 mi 17 RPR A-g-.^s, PNflLYTIC, PSFCinNRL V./Gtt33PL 'rODEL ^Lttl <fcrti<lCA. i£fil<F XIi-WL u. 3. i*Y^ CHART M. Initial Surface Pressure Analysis (Experiment III) 52 5 nCi.--; PROG V./^cijBf^L -f^ODEL FLtET MJif.CPL .tfllMtS CDllRfv U. 5. N0»», Analytic Winds Analytic Winds i )2 HOl^R PROG y/Strdppt '"MODEL FlUT NUf^lCn. t£Pl»CJI CZMliWl U. S. r«)W, CHART N. 6-Hour and 12-Hour Surface Pressure Forecast Using Analytic Winds (Experiment III) 53 Analytic V/lnds Analytic Winds A HOUR PROG Y.^fictj^PL "MODEL fLEii pumica KsiicB cdhrr. u. s. ftf. CHART 0. l8-Hour and 2^-Hour Surface Pressure Forecast Using Analytic Winds (Experiment III) 5^ 't i:>.f-^ ■ 32 nUc^R '^ROG y./£t:Op-fiL '•■MODEL Analytic Winds Analytic Winds \ 36 HOQJR PROG y/£Lti^pPMODEL ,-^\ ftUT lutjiica. ktsKifn :ixim. u. s. •»♦, CHART P. 30-Hour and 36-Hour Surface Pressure Forecast Using Analytic V/inds (Experirnent III) 55 VI. CONCLUSIONS Three cases of analytic data and one case of real data were numerically integrated using a 5-level baroclinic primitive equation model of the general circulation. Experiments v/ere performed using winds derived from the linear balance equation and winds derived analytically. The feasibility of using the linear balance equation to initialize the wind field was examined. In all cases, the forecasts remained meteorological and reasonably well-behaved. The forecasts using winds analytically derived were virtually free of small scale inertial-gravity motions, while the forecasts using winds from the linear balance equation excited intertial-gravity waves which were large and unde- sirable for operational forecasts. Therefore, an important question to be answered is wl it method of balancing should be used to initialize the glol al model so that it will not suffer during the early part o the forecast run from excitation of excessive inertial gravity motions. A number of solutions have been proposed by Myakoda and Moyer (1968), Nitta and Hovermale (1969), and Winninghoff (1971). The method examined by Winninghoff used the equations in the model itself in an iterative sense either at a fixed time level or even in a four dimensional sense in which data is assimilated into a running model. This allows for 56 the natural adjustment mechanism Itself to achieve the desired balance. A time scheme such as the Euler-backward may be used for selective damping of high frequency waves. The iterative method proposed by Winninghoff has the advan- tage of mathematical simplicity and complete consistency with the prediction model. Unfortunately, Winninghoff (1971) estim.ates that the iterative procedure must continue for an equivalent of an I8 to 2k hour forecast. This, of course, is not operationally feasible at the present time. Consequently, the technique which is now being tested by PNV/C utilizes the best vertical mass structure possible obtained by the latest variational analysis techniques available and then solves the complete balance equation at each level in order to get the initial wind fields. This initialization along v/ith other helpful operationally tested devices such as smoothing, turning friction on slowly and not calling on convective adjustment for the first several time steps should be a satis-factory interim solution to this most difficult problem. 57 LIST OF REFERENCES 1. Arakawa, Akio, Katayama, Akira, and Mintz, "Numerical Simulation of the General Circulation of the Atmosphere," Proceedings of the V/MO/IUGG Symposium on Numerical Weather Prediction , Tokyo, Japan, November 25 - December W] 196b, Japan Meteorological Agency, Tokyo, Mar. I969, pp. IV-1 - IV-14. 2. Arakawa, Akio, "Computational Design for Long-Term Numerical Integration of the Equations of Fluid Motion: Two Dimiensional Incompressible Flow. Part I," Journal of Computational Physics , Vol. 1, No. 1, Academic Press, Inc., New York, N.Y., Jan. I966, pp. 119-1^3. 3. Burington and Torrance, Higher Mathematics , p. ^2^- 429, McGraw-Hill, 1939. H. Dickson, Robert R. , and Posey, Julian, "Maps of Snow- Cover Probability for the Northern Hemisphere," Monthly Weather Review , Vol. 95, No. 6, June I965, p. 347-353. 5. Gates, W. L. , "On the Truncation Error, Stability and Convergence of Difference Solutions of the Barotropic Vorticity Equation," J. Meteor. , V. I6, p. 556-568, 1959. 6. Gates, W. L. , and Riegel, C. A., "A Study of Numerical Errors in the Integration of Barotropic Flow on a Spherical Grid," J. of Geoph. Res. , V. 67, No. 2, p. 773-784, Feb. 19^^: 7. Haltiner, G. J., Numerical Weather Prediction , p. 1-39, 90-114, 193-196 and 220-243, Wiley, 1971. 8. Haltiner, G. J., and Martin, F. L. , Dynamical and Physical Meteorology , p. 52-53, McGraw-Hill, 1957. 9. Haurwitz, B. , 1940: The Motion of Atmospheric Distur- bances. J. Marine Research (Sears Foundation) , V. 3, p. 35-50. 10. Haurwitz, B. , 1940: The Motion of Atmospheric Distur- bances on the Spherical Earth. J. Marine Research (Sears Foundation) , V. 3, p. 254-267. 58 11. Heburn, G. W. , Numerical Experiments with Several Time Differencing Schemes with a Barotropic Primitive Equation Model on a Spherical Grid , M. S. Thesis, Naval Postgraduate School, 1972. 12. Jeffreys and Jeffreys, Methods of Mathematical Physics , p. ^29-^31, Cambridge, 1956. 13. Kesel and Winninghoff, "The Fleet Numerical Weather Central Operational Primitive-Equation Model," Mon. Wea. Rev. , V. 100, No. 5, p. 360-373, 1972. 14. Kreyszig, E. , Advanced Engineering Mathematics , p. 175-178, Wiley, 1962. 15. Moyer, R. W. and Miyokoda, K. , "A Method of Initiali- zation for Dynamical Weather Prediction," Tellus , V. 20, p. 115-128, 1968. 16. Naval Postgraduate School Report NPS-51Wu7108lA, Restorative-Iterative Initialization for a Global Prediction Model , by F. J. V/inninghof f , September 1971.- 17. Neam.tan, S. M. , "The Motion of Harmonic Waves in the Atmosphere," J. Meteorology , V. 3, p. 53-56, 19^6. 18. Nitta, T. and Hovermale, J. B. , "A Technique of Objective Analysis and Initialization for the Primitive Forecast Equations," Mon. Wea. Rev. , V. 97, p. 652- 658, 1969. 19. Philips, N. A., "The General Circulation of the Atmosphere: a Numerical Experiment," Quart. J. Meteor. Soc. , V. 82, p. 123-16^, 1956. 20. Rossby, C. G. et al., "Relations Between Variations in the Intensity of the Zonal Circulation of the Atmosphere and the Displacements of the Semi- Permanent Centers of Action," J. Marine Res. (Sears Foundation) , V. 2, p. 38-55, 1939. 21. UCLA Department of Meteorology Technical Report No. 3, Description of the Mint z-Arakawa Numerical General Circulation Model , by W. E. Langlois and C. W. .Kwok, 1959^: 22. Smagorinsky et al. , "Numerical Results From a Nine-Level General Circulation Model of the Atm.osphere , " Monthly Weather Review, Vol. 93, No. 12, Dec. 1965, PP- 727-768. 59 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia 2231^ 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 939^0 3. Dr. George J. Haltiner 5 Chairman, Department of Meteorology Naval Postgraduate School Monterey, California 939^0 ^. Associate Professor Roger T. V/illiams 5 Code 51 Department of Meteorology Naval Postgraduate School Monterey, California 939^0 5. Lieutenant William T. Elias 5 Fleet Numerical Weather Central Naval Postgraduate School Monterey, California 939^0 6. Officer in Charge 1 Environmental Prediction Research Facility Naval Postgraduate School Monterey, California 939^0 7. Commanding Officer 1 Fleet Numerical V/eather Central Naval Postgraduate School Monterey, California 939^0 8. LCDR Wayne R. Lambertson 1 Fleet Numerical Weather Central Naval Postgraduate School Monterey, California 939^0 9. ARCRL - Research Library 1 L. G. Hanscom Field Attn: Nancy Davis/Stop 29 Bedford, Massachusetts 01730 10. Director, Naval Research Laboratory 1 Attn: Tech. Services Info. Officer Washington, D. C. 20390 60 11. American Meteorological Society ^5 Beacon Street Boston, Massachusetts 02128 12. Department of Meteorology Code 51 Naval Postgraduate School Monterey, California 939^0 13. Department of Oceanography Code 58 Naval Postgraduate School Monterey, California 939^0 14. Office of Naval Research Department of the Navy Washington, D. C. 20360 15. Comm.ander, Air Weather Service Military Airlift Command U.S. Air Force Scott Air Force Base, Illinois 62226 16. Atmospheric Sciences Library National Oceanographic Atmospheric Administration Silver Spring, Maryland 20910 17. National Center for Atmospheric Research Box 1470 Boulder, Colorado 80302 18. Dr. T. N. Krishnamurti Department of Meteorology Florida State University Tallahassee, Florida 32306 19. Dr. Fred Shuman Director National Meteorological Center Environmental Science Services Administration Suitland, Maryland 20390 20. Dr. J. Smagorinsky Director Geophysical Fluid Dynamics Laboratory Princeton University Princeton, New Jersey 085^0 21. Dr. A. Arakawa Department of Meteorology UCLA Los Angeles, California 90024 61 22. Professor N. A. Phillips 1 5^-1^22 M. I. T. Cambridge, Massachusetts 02139 23. Dr. Russell Elsberry 1 Department of Meteorology Naval Postgraduate School Monterey, California 939^0 2h. Dr. Jerry D. Mahlman 1 Geophysical Fluid Dynamics Laboratory Princeton University Princeton, New Jersey 085^0 25. Dr. Robert L. Haney 1 Department of Meteorology Naval Postgraduate School Monterey, California 939^0 26. Dr. Ron L. Albertv 1 National Severe Storm Laboratory I6l6 Halley Circle Norman, Oklahoma 27. Dr. W. L. Gates ^ 1 The RAND Corporation 1700 Main Street Santa Monica, California 90H06 28. Dr. Richard Alexander 1 The Rand Corporation 1700 Main Street Santa Monica, California 90406 29. Commanding Officer 1 Fleet Weather Central Box 110 FPO San Francisco 966IO 30. Dr. F. J. Winninghoff . 1 Department of Meteorology UCLA Los Angeles, California 9002^ 31. LCDR P. G. Kesel ' 1 ODSI 2^60 Garden Road Monterey, California 939^0 • 62 32. Mr. Leo C. Clarke PNWC Naval Postgraduate School Monterey, California 939^0 33. Naval Weather Service Command Washington Navy Yard Washington, D. C. 20390 63 Sfcuritv Classification DOCUMENT CONTROL DATA -R&D iSfcurity f f.ss.fic.don o( title, body ot abstract and indexine annotation must be •nitred u-hen the overall ttpott Is cUsslllod) iN*TtfsiG ACTIVITY (Corporate author) Naval Postgraduate School Monterey, California 939^0 2». REPORT SECURITY C L A SSI F I C A T I Oh Unclassified 26. CROUP EPOR T TITLE Numerical Experiments with a Five-Level Global Atmospheric Prediction Model Using a Staggered, Spherical, Sigma Coordinate System ^ ESCRIPTIVE NOTES (Type of report and,lncluaive datet) Master's Thesis; March 1973 U T HO R IS I ff"*' n«me, middle initial, latl name) William Theodore Elias EPORT DATE March 1973 CONTRACT OR GRANT NO. PROJEC T NO. DISTRIBUTION STATEMENT T. TOTAL NO. OF PACES 65 7b. NO. OF REFS 21 Sa. ORIGINATOR'* REPORT NUMBERIS) 86. OTHER REPORT NO(S) (Any other numbmre that may be aaalg^ed thie report) Approved for public release; distribution unlimited. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Postgraduate School Monterey, California 939^0 ABSTRACT Three cases of analytic data and one case of real data wer numerically integrated using a 5-level baroclinic primitive equations model of the general circulation. Experiments were performed using initial winds derived from the linear balance equation and also winds derived analytically. The feasibility of using the linear balance equation to initialize the wind field was examined. In all cases, the forecasts remained meteorological and reasonably well-behaved. Nevertheless, the forecasts derived from initial winds generated by the linear balance equation excited large, operationally-undesirable inertial-gravity waves, while the forecasts from analytically determined initial winds remained free of such small scale "noise". FORM 1/L7'3l (PAGE 1) I NOV *S t *^ / w N 0101 -807-681 1 6i| Security Clacaification A-ai4oa Security Classifirntion KEY yvo ROI Numerical Atmospheric Global Baroclinic Staggered Grid Spherical Coordinates Sigma Coordinate System Primitive Equation ROLE WT DD ,Z\'.A473 'B*^'^: S/N OIOI-807.6S2I 65 Security Cl«B>ification A- 3M09 Thesis :316 c.l U2023 Elias Numerical experiments with a five-level global atmospheric prediction model using a staggered, spherical, sigma coordi- nate system. '>?> NOV 73 21683 / 1 1 Thesis E316 c.l 11+2023 Elias Numeric;^! experiments with a five-level global atmospheric prediction model using a staggered, spherical, sigma coordi- nate system. r^^. thesE316 Numerical experiments with a five-level i 3 2768 001 89278 9 DUDLEY KNOX LIBRARY